Corollary 1

Let G be a complete metrizable group, let G0 be a dense subgroup of G and let H_0 be a closed normal subgroup of G_0. If H is the closure of H_0 in G, the quotient group G_0/H_0 has a completion isomorphic to G/H.*

H is a normal subgroup of G (Chapter III, § 2, no. 3, Proposition 8) and Proposition 4 shows that G/H is complete. Also if $\varphi$ is the canonical mapping of G onto G/H, it is clear that $\varphi(G_0)$ is dense in G/H. The result therefore follows from Chapter III, § 2, no. 7, Proposition 21.

Let G, G' be two Hausdorff abelian topological groups, and $\hat{G}, \hat{G}'$ their respective completions. We recall (Chapter III, § 3, no. 3, Proposition 5) that if $u$ is a continuous homomorphism of G into G', then $u$ is uniformly continuous and extends uniquely to a continuous homomorphism of $\hat{G}$ into $\hat{G}'$, which we shall denote by $\hat{u}$ in the remainder

(*) There exist non-metrizable complete groups G containing a closed subgroup H such that G/H is not complete.

of this subsection. The diagram

in which $i, i'$ are the canonical injections) is commutative. If $v$ is a continuous homomorphism of $G'$ into a Hausdorff abelian topological group $G''$, and if $w = v \circ u$, then it is follows immediately that $\hat{w} = \hat{v} \circ \hat{u}$.

Let $H$ be a closed subgroup of $G$ and let $E = G/H$. Let $j : H \to G$ and $p : G \to E$ be the canonical mappings. Let $\overline{H}$ be the closure of $H$ in $\hat{G}$; $\overline{H}$ is a complete group and we identify it with the completion $\hat{H}$ of $H$. The continuous extension $\hat{j}$ of $j$ to $\hat{H}$ is evidently the canonical injection of $\hat{H}$ in $\hat{G}$.

Suppose from now on that $G$ is metrizable. Then the canonical mapping of $E = GH$ into $\hat{G}\hat{H}$ is a topological isomorphism of $E$ onto a dense subgroup of the complete group $\hat{G}/\hat{H}$ (Corollary 1), and thus we may identify $\hat{G}\hat{H}$ with $\hat{E}$ and the continuous extension $\hat{p}$ of $p$ to $\hat{G}$ with the canonical mapping of $\hat{G}$ onto $\hat{G}/\hat{H}$.